1 (D) A flat metal plate has the shape of the unit disc r² + y² <1. The plate, including the boundary, is heated so that

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1 D A Flat Metal Plate Has The Shape Of The Unit Disc R Y 1 The Plate Including The Boundary Is Heated So That 1 (151.83 KiB) Viewed 57 times

can you please explain how the centre locate at (1/2, 0). thanks
1 (D) A flat metal plate has the shape of the unit disc r² + y² <1. The plate, including the boundary, is heated so that the temperature at any point (x, y) is T(x, y) = x + 2 y2 - x (i) The level curves of T(x, y) are called isothermals because at all points on an isothermal the temperature is the same. Sketch the isothermal T= 0 on the plate together with the boundary of the plate, the gradient and the antigradient at (0,0). (ii) Find the rate of change of temperature at (-1/2, V3/2) in the direction toward the origin (0,0). (11) In which direction does the temperature increase most rapidly at (0, 0)? In which direction does it decrease most rapidly at (0, 0)? Specify these directions by unit vectors. What is the maximal rate of increase? What is the maximal rate of decrease? (iv) Find the coldest point in the interior of the plate and the temperature at this point,